A367907 \/ U A367908 \/ U A367909 = A000027.

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editing

editing

proposed

A367907 \/ A367908 \/ A367909 = A000027.

BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in exactly Numbers n such that there is only one way to choose a different binary index of each binary index of n.

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in exactly one way.

Affirmations are If there is at least one choice we get A367906, contradictions A367907counted by A367902.

If there are multiple ways no choices we get A367909A367907, counted by A367903.

If there are multiple choices we get A367909, counted by A367772.

A291166 lists numbers whose binary indices are relatively prime.

A368098 counts non-isomorphic unlabeled multiset partitions satisfying the for axiom, complement A368097.

Cf. A000612, A055621, A059519, A072639, A083323, A309326, `A326032, A326675, A326702, A326753, A326872, `A330226, A355529, `A355740, A368100A367902, A367912.

Cf. `A367769, `A367770, `A367901, A367902, A367903, A367912, A367913, A367915.

allocated for Gus WisemanBII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in exactly one way.

1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 56, 67, 69, 70, 73, 74, 81, 88, 98, 104, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152, 154, 156, 162, 163, 165, 166, 168

1,2

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary digits (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.

The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in exactly one way, namely (1,2,3), so 21 is in the sequence.

The terms together with the corresponding set-systems begin:

1: {{1}}

2: {{2}}

3: {{1},{2}}

5: {{1},{1,2}}

6: {{2},{1,2}}

8: {{3}}

9: {{1},{3}}

10: {{2},{3}}

11: {{1},{2},{3}}

13: {{1},{1,2},{3}}

14: {{2},{1,2},{3}}

17: {{1},{1,3}}

19: {{1},{2},{1,3}}

21: {{1},{1,2},{1,3}}

22: {{2},{1,2},{1,3}}

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Select[Range[100], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]==1&]

These set-systems are counted by A367904, non-isomorphic A368099.

Positions of 1's in A367905, firsts A367910, sorted firsts A367911.

Affirmations are A367906, contradictions A367907.

If there are multiple ways we get A367909.

The version for MM-numbers of multiset partitions is A368101.

A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A059201 counts covering T_0 set-systems.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

A291166 lists numbers whose binary indices are relatively prime.

A326031 gives weight of the set-system with BII-number n.

A368098 counts non-isomorphic multiset partitions satisfying the axiom, complement A368097.

Cf. A000612, A055621, A059519, A072639, A083323, A309326, `A326032, A326675, A326702, A326753, A326872, `A330226, A355529, `A355740, A368100.

Cf. `A367769, `A367770, `A367901, A367902, A367903, A367912, A367913, A367915.

BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers), A326783 (uniform), A326784 (regular), A326788 (simple), A330217 (achiral).

allocated

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Gus Wiseman, Dec 11 2023

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editing

allocated for Gus Wiseman

allocated

approved